Koornwinder polynomials

In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder and I. G. Macdonald, that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C^∨
_n, C_n), and in particular satisfy analogues of Macdonald's conjectures. In addition Jan Felipe van Diejen showed that the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them. Furthermore, there is a large class of interesting families of multivariable orthogonal polynomials associated with classical root systems which are degenerate cases of the Macdonald-Koornwinder polynomials. The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.

The Macdonald-Koornwinder polynomial in n variables associated to the partition λ is the unique Laurent polynomial invariant under permutation and inversion of variables, with leading monomial x^λ, and orthogonal with respect to the density

$\prod _{1\leq i<j\leq n}{\frac {(x_{i}x_{j},x_{i}/x_{j},x_{j}/x_{i},1/x_{i}x_{j};q)_{\infty }}{(tx_{i}x_{j},tx_{i}/x_{j},tx_{j}/x_{i},t/x_{i}x_{j};q)_{\infty }}}\prod _{1\leq i\leq n}{\frac {(x_{i}^{2},1/x_{i}^{2};q)_{\infty }}{(ax_{i},a/x_{i},bx_{i},b/x_{i},cx_{i},c/x_{i},dx_{i},d/x_{i};q)_{\infty }}}$

on the unit torus

$|x_{1}|=|x_{2}|=\cdots |x_{n}|=1$,

where the parameters satisfy the constraints

$|a|,|b|,|c|,|d|,|q|,|t|<1,$

and (x;q)_∞ denotes the infinite q-Pochhammer symbol. Here leading monomial x^λ means that μ≤λ for all terms x^μ with nonzero coefficient, where μ≤λ if and only if μ₁≤λ₁, μ₁+μ₂≤λ₁+λ₂, …, μ₁+…+μ_n≤λ₁+…+λ_n. Under further constraints that q and t are real and that a, b, c, d are real or, if complex, occur in conjugate pairs, the given density is positive.

References

• Koornwinder, Tom H. (1992), "Askey-Wilson polynomials for root systems of type BC", Contemporary Mathematics, 138: 189–204, doi:10.1090/conm/138/1199128, MR 1199128, S2CID 14028685
• van Diejen, Jan F. (1996), "Self-dual Koornwinder-Macdonald polynomials", Inventiones Mathematicae, 126 (2): 319–339, arXiv:q-alg/9507033, Bibcode:1996InMat.126..319V, doi:10.1007/s002220050102, MR 1411136, S2CID 17405644
• Sahi, S. (1999), "Nonsymmetric Koornwinder polynomials and duality", Annals of Mathematics, Second Series, 150 (1): 267–282, arXiv:q-alg/9710032, doi:10.2307/121102, JSTOR 121102, MR 1715325, S2CID 8958999
• van Diejen, Jan F. (1995), "Commuting difference operators with polynomial eigenfunctions", Compositio Mathematica, 95: 183–233, arXiv:funct-an/9306002, MR 1313873
• van Diejen, Jan F. (1999), "Properties of some families of hypergeometric orthogonal polynomials in several variables", Trans. Amer. Math. Soc., 351: 233–70, arXiv:q-alg/9604004, doi:10.1090/S0002-9947-99-02000-0, MR 1433128, S2CID 16214156
• Noumi, M. (1995), "Macdonald-Koornwinder polynomials and affine Hecke rings", Various Aspects of Hypergeometric Functions, Surikaisekikenkyusho Kokyuroku (in Japanese), vol. 919, pp. 44–55, MR 1388325
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9, MR 1976581
• Stokman, Jasper V. (2004), "Lecture notes on Koornwinder polynomials", Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Hauppauge, NY: Nova Science Publishers, pp. 145–207, MR 2085855